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Portfolio Rebalancing — Common Misconceptions John Huss Global Asset Allocation February 2017 Decisions relating to portfolio rebalancing, while Thomas Maloney often considered secondary to deciding on the Portfolio Solutions Group allocations themselves, can be considered an active investment strategy and have important implications for expected (and realized) portfolio returns and risk. In this article we address common misconceptions about the role and implications of rebalancing, particularly in the context of actively-managed portfolios. These include the so-called “rebalancing premium” and the impact of rebalancing on the expected performance of risk-targeted and levered portfolios. A companion article (Ilmanen and Maloney (2015)) examines the rebalancing of strategic asset portfolios. We thank Jeff Dunn, Jeremy Getson, Antti Ilmanen, Ronen Israel, Chris Palazzolo, Lasse Pedersen, Scott Richardson, Kari Sigurdsson, Rodney Sullivan and Ashwin Thapar for helpful comments and suggestions. AQR Capital Management, LLC Two Greenwich Plaza Greenwich, CT 06830 p: 203.742.3600 f : 203.742.3100 www.aqr.com Portfolio Rebalancing – Common Misconceptions Executive Summary 1 instruments can be suitable for long-term Rebalancing might seem at first glance to be just investors, subject to their risk preferences. a necessary chore of portfolio management, but it An appendix covers several additional topics, is nonetheless a topic surrounded by controversy. including Inconsistent rebalancing processes applied to a dynamic use of terminology and varied interpretations of the underlying math have spread more confusion than understanding, and anecdotes are too often presented as definitive data. In this article we refute four prevalent misconceptions related to portfolio rebalancing and seek to clarify the practical implications of each of these topics. Our conclusions may be summarized as follows: 1. Rebalancing is not generally, as some outperformance. It does alter the distribution of possible return outcomes for a portfolio, but this is more correctly and usefully interpreted as a risk-reduction effect from maintaining better diversification. rebalancing allocations simple illustration of several portfolio.1 Misconception 1: “Rebalancing is a Source of Smart Beta Outperformance” Some smart beta managers have suggested that periodically rebalancing a portfolio to its target weights is itself a source of returns,2 and even the main source of expected outperformance. Rebalancing is said to “harvest a premium” that does not require mean reversion in prices. have suggested, a source of smart beta 2. While a to maintains constant long-term capital risk characteristics better than ‘buy and hold’, dynamic risk targeting does an even better job. 3. There is no reliable evidence that risk-targeted portfolios suffer a drag on returns from selling ‘after the horse has bolted.’ Evidence of the risk benefits, on the other hand, is pervasive. Fact: Rebalancing does not earn positive returns – and is not a positive expected return strategy – unless prices are mean-reverting at a frequency the rebalancing process can capture. It does, however, help to maintain diversification and so alter the distribution of possible outcomes for the portfolio. The claim that rebalancing can “harvest a premium” even in the absence of mean reversion (and regardless of the target weights) is based on a stretched interpretation of the math, as we illustrate below. Details It is a curious fact that a periodically rebalanced portfolio earns a positive return (“turns water into additional wine”) if the component assets all end up at their rebalancing process, but this does not cause starting prices.3 The resolution of this seeming any special “drag” effects beyond the normal paradox is that such an outcome already implies compounding effects of investments at a similar some mean reversion (the assets have reverted to risk level, and some additional transaction their starting prices). It also requires the assets to costs. Moderately levered portfolios of liquid have positive arithmetic mean (AM) returns.4 4. Levered portfolios require an 1 This article focuses on rebalancing in response to inputs other than changing investment signals or expected returns. The question of how and when to apply new investment signals – weighing the costs of holding a sub-optimal portfolio against the costs of trading to new targets – is highly dependent on the strategy and its investment universe. 2 Based on an extensive literature on “diversification return” perhaps starting with Booth and Fama (1992) and including Bernstein and Wilkinson (1997), Fernholtz (2002) and Willenbrock (2010). See Ilmanen (2011) for a concise overview. For the smart beta context, see for example Azuelos and Yasenchak (2014). For an analytical framework that attributes relative performance of rebalanced portfolios to two opposing components, see Hallerbach (2014). 3 Erb and Harvey (2006) use commodity futures as an example. 4 AM is always higher than GM (where volatility is non-zero), and in this case it is the assets’ GM returns that are zero. An asset whose price rises from 2 Portfolio Rebalancing – Common Misconceptions In the above case, the outperformance generated For these reasons, rebalancing tends to narrow by rebalancing is equal to the “diversification the distribution of terminal wealth outcomes for return,” usually defined as the difference between the portfolio and make it less positively-skewed. the geometric mean return (GM) of a portfolio and If all the assets have equal expected returns, the weighted average GM of its components. But rebalancing increases the median wealth outcome this equivalence does not usually hold. Indeed, while leaving the mean unchanged: Exhibit 1 the practical relevance of “diversification return” illustrates this using simulated data. (In the case is limited by the fact that in general the weighted of, say, a portfolio of stock and bond allocations average GM is not the return of any investable with dissimilar expected returns, rebalancing portfolio. The buy-and-hold portfolio, which is actually reduces the mean expected terminal investable, may have a higher or lower GM than wealth by preventing dominance of the higher- the portfolio rebalanced to constant weights. return asset, but the narrowing and reshaping 5 While diversification return should not be effect is even more pronounced.) considered a return premium, we believe it does highlight why AM and GM rates of return are both incomplete measures of expectation which are Exhibit 1 ‒ Simulated Impact of Rebalancing on Terminal Wealth (TW) Outcomes (10-Year Horizon) better understood in the context of the distribution Measure of terminal wealth outcomes. Mean TW $2.71 $2.71 Same mean Median TW $2.51 $2.54 Higher median Rebalancing reshapes the distribution of terminal 90th percentile $4.11 $4.05 Lower best cases wealth outcomes, by neutralizing compounding 10th percentile $1.55 $1.59 Higher worst cases effects within the portfolio. It prevents winning 90th-10th Range $2.56 $2.46 Narrower range 6 investments from earning higher weights and losers from decaying to lower weights. In other words, a rebalanced portfolio forgoes the very best buy-and-hold outcomes (the right tail of the distribution), where winning investments keep on Buy & Hold Rebalance Impact Volatility 12.1% 11.5% Lower volatility Max Drawdown -18.3% -16.9% Smaller drawdowns Source AQR. 50,000 simulations based on the assumption of a $1 portfolio of three uncorrelated assets with constant expected volatility of 20%, expected arithmetic Sharpe ratio 0.5, normally-distributed serially-independent returns, and daily rebalancing. Gross of transaction costs. winning and compounding their gains, and losers fizzle out to small, inconsequential weights. But it compensates by outperforming in many other outcomes where performance across investments is less divergent.7 Importantly, rebalancing also tends to maintain a lower risk-level than buyand-hold by preventing the concentration of risk among winning investments. This change in the shape of the distribution of terminal wealth means that a rebalanced portfolio is more likely to realize positive returns – and more likely to realize returns equal to or exceeding the mean – over the investment horizon. These are risk-related qualities that many investors would prefer in their portfolios, so it is no surprise that most do choose to periodically rebalance. $100 to $110 (+10%) and then falls back to $100 (-9%) has a positive arithmetic mean return. 5 E ven a rebalanced portfolio of assets which follow a true random walk is associated with a positive diversification return, and yet in this hypothetical situation it is not possible to apply investment skill or “harvest a premium” as the results are purely random. 6 For a discussion of return estimation, see for example Jacquier, Kane and Marcus (2003), Hughson, Stutzer and Yung (2006), and references therein. Some recent articles (for example Chambers and Zdanowicz (2014)) have questioned the usefulness of rates of return in general and especially GM returns, preferring to focus solely on mean expected terminal return or terminal wealth (TW). We have sympathy for this position but believe it is important to consider the distribution of possible outcomes, and not just the mean. We also emphasize that the use of terminal return assumes the investor can ‘stay the course’ regardless of the path of returns. Qian (2014) also considers the impact of rebalancing on terminal wealth distributions, and defines a “wealth Sharpe ratio” as mean excess TW / standard deviation of TW. 7 See Wise (1996) for an analysis of the probability of outperformance. Portfolio Rebalancing – Common Misconceptions Some authors have interpreted these effects as a return premium for buying losers (effectively selling portfolio insurance) and forgoing the best outcomes by selling winners, i.e., applying a strategy which itself has a negatively-skewed return distribution.8 But we believe it is more useful (and mathematically correct) to describe these effects as the risk-reduction benefits of maintaining better diversification. Contrarian rebalancing does not harvest returns in the sense of an increased mean expected wealth,9 unless prices tend to mean-revert at a frequency the rebalancing process can capture.10 Rebalancing also incurs costs, which has implications for smart 3 Details Risk rebalancing is the practice of rebalancing to target risk allocations, rather than capital allocations, based on dynamic risk estimates for each investment. So long as the risk estimates don’t change, this is exactly the same contrarian process as capital rebalancing – selling winners and buying losers – with the same tendency to outperform if investments experience meanreverting performance, and the same impact on the distribution of terminal wealth outcomes (raising the median and reducing the positive skew, as detailed above). beta portfolios as we discuss in the appendix. When risk estimates do change, some additional Misconception 2: “Constant Capital Equals Constant Risk” rebalancing will be required to maintain risk While investors may not explicitly assert the time-varying with some predictability. A risk- above, traditional rebalancing processes that rebalanced portfolio may therefore maintain its aim to maintain benchmark capital weights (for diversification and risk characteristics better than example, 60% stocks and 40% bonds) do imply a a capital-rebalanced portfolio. allocations. Below we provide empirical evidence that many investments’ risk characteristics are belief that capital weights determine the relevant risk characteristics of a portfolio. Exhibit 2 shows realized volatility contributions through time for three stock/bond portfolios over Fact: The riskiness of assets varies over time, and a 41-year period. Portfolio (a) allocates 20% to the future riskiness of assets is easier to predict than equities and 80% to bonds at the start of the period, their future returns. While rebalancing to constant and does not rebalance. This portfolio’s volatility capital allocations does indeed help to maintain gradually becomes dominated by equities. Portfolio long-term risk characteristics (see Ilmanen and (b) starts with the same allocation and rebalances Maloney (2015)), dynamic risk targeting does an to maintain it. This portfolio achieves a more even better job. Seemingly passive buy-and-hold consistent risk allocation, though changing asset portfolios tend to have the most variable and least volatilities cause significant variations. Portfolio predictable risk outcomes. (c) allocates by risk, not capital, and rebalances to 11 maintain equal risk allocations through time. It achieves the most consistent risk contributions, by 8 As described by Perold and Sharpe (1988), Harvey et al (2014) and Hillion (2016). Harvey et al suggest a rebalanced portfolio is susceptible to larger drawdowns than a buy-and-hold portfolio (in contrast to the results in Exhibit 1 above). This is true for two portfolios entering a period of sustained investment losses with the same weights, as the authors illustrate. However, this analysis misses the tendency of rebalanced portfolios to be already better diversified at the onset of such periods. When considered in the context of longer investment horizons, as above, rebalanced portfolios tend to experience smaller drawdowns because of their lower average risk level. 9 As emphasized by Chambers and Zdanowicz (2014). Mean expected wealth may be the most relevant measure of expectation when considering components of a wider portfolio. For example, if the investments’ next-day returns were negatively correlated to their past-month returns, a portfolio rebalanced to fixed weights daily or monthly would capture a larger positive expected return than one rebalanced annually. 10For example, if the investments’ next-day returns were negatively correlated to their past-month returns, a portfolio rebalanced to fixed weights daily or monthly would capture a larger positive expected return than one rebalanced annually. 11A result formalized and brought to wide attention by Engle (1982) but dating back many decades. 4 Portfolio Rebalancing – Common Misconceptions adjusting target weights based on recent relative therefore narrowing the range of outcomes gives asset volatilities. investors more scope to seek higher returns on average. It also increases diversification over Exhibit 2 ‒ Realized Volatility Contributions for Three Rebalancing Approaches, 1975-2015 Exhibit 3 compares the rolling 60-day volatility 100% of a constant-dollar investment in U.S. equities 75% to that of a simple risk-targeting strategy based 50% on past 60-day volatility. The risk target is set to match the full-sample constant-dollar volatility 25% 0% Equities Bonds Exhibit 3 ‒ Realized Volatility of U.S. Equities, 1975-2015 Contribution to 60d Volatility 100% 75% 90% 50% 25% 0% of 18% for visual comparison. The volatility of volatility is reduced by 54%. B. Capital Rebalanced Equities Bonds C. Risk Rebalanced 80% 70% 60% 50% 40% 30% 20% 10% 100% Contribution to 60d Volatility dominating long-term performance. Rolling 60-day Volatility Contribution to 60d Volatility A. Buy and Hold time, reducing the likelihood of short periods 0% 1975 75% 1980 1990 1995 Constant Dollar 50% 25% 1985 2000 2005 2010 2015 Risk-Targeted Source: AQR. Based on daily returns. Equities Bonds 0% The benefits of RT are not only found in equities. Source: AQR. Starting capital allocations for (a) and (b) are 20% equities, 80% bonds. Based on daily returns of U.S. equities and U.S. 10-year Treasuries, and daily rebalancing. Risk estimates for risk-rebalanced portfolio are based on rolling 60-day volatility. Exhibit 4 shows the corresponding reduction in volatility of volatility for 34 assets, using the same simple strategy. Reductions vary from 37% to 65%, with the amount of reduction achieved being What if all of the portfolio’s components become more risky? Risk-targeted (RT) portfolios go one step further than risk-rebalancing by scaling each investment individually and allowing variations in the total capital invested, rather than merely adjusting relative allocations. The main objective of risk-targeting is to provide more stable realized volatility. A portfolio should be designed with the worst case in mind, and closely related to the persistence or autocorrelation of volatility for that asset. RT works when volatility is persistent, and volatility is persistent everywhere we look. Although even naïve risk-targeting can produce more stable realized volatility, all risk targeting processes are not equal. Each depends on a risk model, consisting of estimates of the volatility of each asset and of correlations between them. Estimates are typically based primarily on Portfolio Rebalancing – Common Misconceptions 5 Exhibit 4 ‒ Volatility Persistence and the Impact of Risk-Targeting, 1975-2015 EQUITIES BONDS COMMODITIES TIPS 0.8 0.6 0.4 0.2 Autocorrelation of 60d Volatility UK 10y US 10y Wheat Soybeans Silver Natural Gas Gold Heating Oil Corn Crude Copper BCOM Index UK 10y GSCI Index Japan 10y Germany 10y US 10y Australia Sweden Hong Kong Italy Norway Spain Netherlands France Germany UK Switzerland Japan Canada NASDAQ RU2K Dow Jones S&P 500 0.0 % Reduction in Vol-of-vol from Risk-Targeting Source AQR. Hypothetical strategies based on daily returns from 1975 where available; some series start later. Based on daily rebalancing, gross of transaction costs and fees. Hypothetical data have inherent limitations, some of which are disclosed in the appendix. historical returns and must balance responsiveness risk/return interaction is mixed and depends with the need to minimize estimation noise and on the investment, time period and method unnecessary turnover. for estimating risk. Some of the literature has The portfolio volatility target is usually a constant number corresponding to the investor’s risk appetite.12 The strategy may target this constant risk level at every rebalance, or, more likely, it may permit short-term variation in the target depending on investment signals, constraints and other risk controls, but be calibrated with the aim of achieving the target over the long-term. contrarian-minded investors benefits of risk-targeting – more stable volatility outcomes – are supported by clear evidence across many investments. Details Risk-targeted (RT) portfolios sell investments when they become more volatile. Therefore if different levels of recent volatility predict higher Misconception 3: “Risk-Targeted Portfolios Sell When Assets are Cheap and Suffer a Drag on Returns” Some misinterpreted this evidence. By contrast, the risk assume that risk-targeted strategies are doomed to sell investments too late, ‘after the horse has bolted.’ or lower subsequent returns on average, the RT process may have an impact on expected returns. The myth stated above implies two separate assumptions about this relationship, both commonly held but both requiring scrutiny. Indeed it has been suggested that for patient long- The first is that RT portfolios sell investments after term investors, spikes in volatility are more likely they fall in price. This is true only if increasing to signal a good time to buy than to sell. Some volatility coincides with falling prices. While research has claimed to support the idea that this evidence suggests that equities do indeed exhibit causes a drag on risk-targeted returns. this well-known tendency, other asset classes such Fact: We find no reliable evidence that risk targeting impairs risk-adjusted returns – and for as bonds and commodities do not (see Exhibit A2 in the appendix). a risk-targeted investment, risk-adjusted returns The second questionable assumption is that falling are what matter. The empirical evidence for any prices imply higher future returns (or Sharpe ratios). 12Andersen, Bianchi and Goldberg (2014) also consider time-varying risk targets, e.g., based on the realized volatility of a benchmark portfolio. The practical usefulness of this approach is limited, given that the main reason for risk targeting is that investor risk tolerance does not increase with market volatility. 6 Portfolio Rebalancing – Common Misconceptions EQUITIES 0.2 BONDS COMMODITIES TIPS 0.1 0.0 UK 10y US 10y Wheat Soybeans Silver Heating Oil Natural Gas Gold Crude Corn Copper BCOM Index UK 10y GSCI Index Germany 10y US 10y Japan 10y Australia Hong Kong Sweden Italy Norway Spain Netherlands France Germany UK Switzerland Japan Canada NASDAQ RU2K S&P 500 -0.1 Dow Jones Gross Sharpe Ratio Impact Exhibit 5 ‒ Gross Impact on Sharpe Ratio from Simple Risk-Targeting, 1975-2015 Source AQR. Hypothetical strategies based on daily returns from 1975 where available; some series start later. Based on daily rebalancing, gross of transaction costs and fees. Hypothetical data have inherent limitations, some of which are disclosed in the appendix. The well-documented success of trend-following expect an investment to earn a constant Sharpe strategies, which are based on the opposite belief, ratio through time, with periods of elevated risk already hints that asset price behavior is not so compensated by higher returns? Or are variations simple. Both reversal and momentum effects are in expected returns likely to be unrelated to risk, observed in many asset classes, tending to operate implying a lower prospective Sharpe ratio during at different time horizons, which may explain volatile periods?15 why empirical evidence of the return impact of RT (based on different estimation horizons) has tended to be mixed.13 Moreover, value effects may be better harnessed by cross-sectional strategies than timing strategies.14 Whatever our expectations, it is important to recognize that while time-varying volatility has a predictable component, it also has an unpredictable component. This is why variations in realized volatility can be significantly reduced Exhibit 5 shows the gross Sharpe ratio impact for but the same simple risk-targeting strategy used in unpredictable element to dilute any ex ante Sharpe Exhibit 4. The impact on Sharpe ratios is much ratio impact implied by the above assumptions. less consistent than the impact on the stability of For a risk-targeted portfolio of diversifying assets realized volatility. For this particular strategy and or strategies, it may be reasonable to assume a sample period, the Sharpe ratio impact tends to be very modest long-term Sharpe ratio advantage positive in equities (the S&P 500 being a notable due to improved diversification through time exception), and mixed in other asset classes. In all and across the portfolio: RT is after all primarily cases the impact is modest (-0.1 to +0.2). a risk management strategy, not a return-seeking Not only is the empirical evidence mixed, but ex ante expectations are unclear. Should we not eliminated. We would expect this strategy. Finally, it is important to note that risk-targeting 13Andersen, Bianchi and Goldberg (2014) suggest that a risk-targeted stock-bond risk parity portfolio would have significantly underperformed a fixed-leverage portfolio over a long period, due to a detrimental interaction between the RT portfolio’s leverage (related to recent volatility) and its subsequent returns. However, this was a misinterpretation of their own results (see Asness, Hood and Huss (2015) for details). Other studies, based on other periods and other assumptions, have reported the opposite effect, especially in equities (see for example Perchet et al (2014) and Moreira and Muir (2016)). 14See Asness, Ilmanen and Maloney (2016). 15Both of these scenarios imply a positive Sharpe ratio impact from risk targeting, through mild time diversification in the first case and favorable timing in the second. However, any such benefits would be diluted by (1) the fact that we can forecast risk only imperfectly and (2) transaction costs. A third scenario, where elevated risk is associated with a higher Sharpe ratio, implies a negative impact but is the least well-supported by empirical evidence. Portfolio Rebalancing – Common Misconceptions may NAV (i.e., does not reinvest gains and losses, therefore be most appropriate for the most liquid trading and incurs costs. It but exchanges them with an external pool of instruments and for managers with cost-effective capital). The compounding portfolio outperforms execution infrastructure including cost-optimized during periods of persistent positive or negative rebalancing and patient trading algorithms. performance (the green lines show one such Misconception 4: “Levered Portfolios Underperform because of Variance Drag Associated with Rebalancing, and are Unsuitable for Long-Term Investors Some commentators have suggested that leverage – which can be used by managers to offer the same strategy at different risk levels – incurs a “variance drag” or a “negative rebalancing return” that makes such levered portfolios unsuitable for longterm investors. Fact: Levered portfolios can outperform or underperform the scaled performance of their underlying unlevered reference portfolio, due to compounding effects that depend on the investment outcome. Variance drag, as most commonly discussed, is a misleading concept. Many moderately levered portfolios are suitable for long-term investors, subject to their risk preferences. Details To understand the implications of rebalancing levered portfolios, we must first be clear on the effects of compounding. Most portfolios, whether fully-invested or risk-targeted, are allowed to compound gains and losses. Mathematically, this equates to an absence of contrarian rebalancing at the portfolio level – profits are not withdrawn but reinvested, and losses are not replaced. outcome), but lags during choppy, mean-reverting performance (purple lines). Compounding also, as we mentioned previously, creates a positivelyskewed distribution of outcomes for terminal wealth. Exhibit 6 ‒ Effects of Compounding on Trending and Mean-Reverting Investment Outcomes Cumulative Returns requires 7 Time Trending Outcome, Compounded Trending Outcome, Rebalanced Mean-Reverting Outcome, Compounded Mean-Reverting Outcome, Rebalanced Source AQR. One-year simulations of daily returns with zero expected autocorrelation and 15% expected volatility. The rebalanced investments rebalance to their starting value daily, i.e., they do not reinvest gains and losses. Simulated returns for illustrative purposes only. There are two main approaches to using leverage. One is to explicitly target a leverage ratio (many levered ETFs do this). The other is used by risktargeted strategies, where some variable amount of leverage is required to meet the risk target. The two approaches have different objectives and very different risk characteristics, but they both require an additional rebalancing process that is specific to levered strategies. If the net value of the portfolio changes significantly, some rebalancing will be required to regain the leverage Exhibit 6 compares the simulated performance or risk target. Positive returns will cause the of a typical compounding portfolio with that portfolio to fall below its target risk level, and the of a portfolio that rebalances to a constant manager must buy more investments to regain it.16 16Consider a hypothetical $100 invested in a strategy with a volatility target of 10%, which it initially achieves by holding levered investments totaling $200 in assets with 5% forecast volatility. The strategy makes a 10% gain, while the market risk forecasts remain unchanged. The portfolio volatility 8 Portfolio Rebalancing – Common Misconceptions Similarly, after negative returns the manager must liquid instruments and employing cost-efficient sell investments. trading processes — can indeed be suitable for long- Unlike contrarian rebalancing between investments to maintain portfolio weights (which levered portfolios must also do, unless they are cap-weighted), this additional process has a momentum bias. It has been characterized in some research17 as “incurring a negative diversification return.” However, it is more meaningful to say that it reproduces the compounding effects experienced by an unlevered investment at a similar risk level.18 As described above, these compounding effects can have a positive or negative impact depending on the price path: a 3x levered ETF earns (gross) less than three times the index return during a choppy year, but it outperforms three times the index return during a (positive or negative) trending year. So gross term investors, subject to their risk preferences. Concluding Thoughts Well-managed dynamic rebalancing processes may lead to more predictable risk characteristics, while seemingly passive buy-and-hold portfolios may have the most variable and least predictable risk outcomes. The most dynamic portfolios require a combination of several separate rebalancing processes, each of which has its own rationale and its own effects on risk and return expectations (see appendix for an illustration). In general, rebalancing does not “harvest a return premium,” but it does help to maintain diversification and thereby change the distribution of possible wealth outcomes for a portfolio. of costs and ignoring potential differences in interventions by risk managers or counterparties,19 levered portfolios do not suffer any special “drag” effects beyond the normal compounding effects of investments at a similar risk level. Furthermore, these compounding effects do not reduce the mean expected terminal wealth, unless investment performance is assumed to be mean-reverting. They do, however, require additional turnover and incur transaction costs, which has led some to question their suitability for long-term investors. A levered ETF offering 2x or 3x exposure to an equity index will have a very high risk level (30-50% annual volatility), and will indeed require very active rebalancing. But for a risk-targeted strategy with a volatility of 1020% this effect (and associated transaction costs) is milder, since it scales with the square of the volatility. Such strategies — especially those using The bottom-line return impact over any given investment period will depend on the price paths of investments during that period. Rebalancing topics will no doubt continue to attract research and commentary, and there are often several possible interpretations of the same underlying math or empirical evidence. In this article we have challenged several misconceptions that we believe have arisen from unhelpful or erroneous interpretations, and aimed to substitute more convincing and intuitive explanations. Rebalancing is an essential part of all active investment management. Once the implications have been clearly understood and the most efficient processes implemented, managers and investors can then turn their attention back to the underlying strategy, which is the real source of expected returns. is now approximately 5%*$210 = $10.50,which is only 9.5% of the new NAV of $110. The manager must invest an additional $10 to regain the 10% risk target. The same mechanism applies to portfolios explicitly targeting a leverage ratio. 17See for example Qian (2012). 18To see this, we can compare the example in footnote 15 to an unlevered $100 investment with a volatility of 10% (i.e., $10). After a 10% gain, the volatility of the unlevered investment is $110*10%=$11. This investment achieves by natural compounding the same new risk level that the levered strategy achieves by adding investments 19Strategies levered to target higher risk levels may be subject to relatively tighter risk controls, as tolerance for large drawdowns does not necessarily scale with volatility; these controls may affect returns if they are triggered. Portfolio Rebalancing – Common Misconceptions 9 Appendices Putting It All Together: An Illustration In this article we discussed misconceptions surrounding several different types of rebalancing processes, each being an adjustment to portfolio weights that is made in response to market developments, separate from any views on expected returns. A levered, risk-targeted risk parity strategy is an example of a portfolio that employs all of these processes, though of course it trades only their net sum. In Exhibit A1 we present a simplified illustration of how the processes may be combined. The notes in the last column describe what is happening at each stage. Table (a) shows the starting portfolio, with equal risk allocations to two assets that might represent stocks and bonds (expected volatilities of 15% and 5%, expected correlation zero). To reach its risk target of 10%, the portfolio is approximately 1.9x levered. Exhibit A1 ‒ Worked Example of Rebalancing a Levered, Risk-Targeted Portfolio a. Starting Portfolio Price Asset A Asset B Fund Notes $100.00 $100.00 $100.00 Exposure $47.14 $141.42 $188.56 Fund is 1.89x levered Risk Estimate 15.0% 5.0% Risk Allocation 50.0% 50.0% b. Portfolio After Market Move 10.0% Risk target is 10% 100.0% Equal risk allocation Asset A Asset B Price $90.00 $105.00 $102.36 Asset A -10%, asset B +5%, portfolio +2.4% Exposure $42.43 $148.49 $190.92 Fund now 1.87x levered Risk Estimate 16.0% 5.0% Risk Allocation 47.8% 52.2% Asset A Asset B Fund $5.30 ($5.30) ($0.00) Contrarian shift from winner B to loser A c. Required Rebalancing 1. Rebalance From Winners to Losers Fund 9.8% Notes Asset A now riskier, portfolio below target 100.0% Allocations have shifted Notes 2. Rescale Portfolio Due to Leverage $0.52 $1.57 $2.09 Scale portfolio due to change in NAV 3. Rebalance From Riskier to Less Risky ($2.30) $2.30 ($0.00) Rebalance due to change in risk estimates 4. Rescale Portfolio Due to Risk Changes ($0.72) ($2.30) ($3.02) Scale portfolio due change in risk estimates TOTAL REBALANCE $2.81 ($3.74) ($0.93) Net trades Asset A Asset B Exposure d. Portfolio After Rebalance $45.24 $144.75 Risk Estimate 16.0% 5.0% Risk Allocation 50.0% 50.0% Source AQR. For illustrative purposes only. Fund Notes $189.99 Fund now 1.86x levered 10.0% Portfolio back at 10% risk target 100.0% Equal risk allocations re-established 10 Portfolio Rebalancing – Common Misconceptions Table (b) shows the impact of price changes and changes in risk estimates on the allocations and portfolio risk estimate. Asset A is down 10% and its risk estimate has increased, while asset B is up 5% and its risk estimate is unchanged. The allocations and portfolio risk level have moved away from their targets. Table (c) shows the four different rebalancing processes that are required to regain the target allocations and risk level. These are the processes we have discussed in this article: 1. contrarian rebalancing from winners (asset B) to losers (asset A); 2. rescaling of the levered portfolio due to the change in NAV (all else equal, the increase in NAV had left the portfolio slightly underinvested); 3. risk-rebalancing due to changes in risk estimates (asset A is now riskier); and 4. rescaling of the portfolio due to changes in risk estimates (this and the previous process may be considered part of the same risk-targeting process). Table (d) shows the portfolio after the net rebalancing trades: it has regained its target risk allocations and target portfolio risk level. Additional Implications We conclude by briefly covering two topics that follow from the discussions in the main article: (1) If smart beta outperformance is not directly attributable to rebalancing, is it simply a consequence of weighting by anything other than market capitalization? (2) Could risk targeting exacerbate market volatility by selling at times of market stress? Smart Beta: Anything But Cap-Weighted? Many non-cap-weighted portfolios outperform on paper, gross of costs, but all (unlike the benchmark) require rebalancing and many are not implementable. Below we explain why we believe the best “smart beta” portfolios cost-efficiently implement the highest-expected-return tilts away from the market, including value, momentum and quality. The classic (if imperfect) example of a buy-and-hold portfolio is a capitalization-weighted equity portfolio. This does not require rebalancing except for periodic adjustments to account for issuance, dividends, buy-backs and constituent changes. Cap-weighted portfolios have intuitive advantages relating to costefficiency, representativeness and macro-consistency. They also suffer from well-known disadvantages such as a tendency to overweight (and become dominated by) overvalued stocks, sectors or countries. Arnott et al (2013) showed that many non-cap-weighted portfolios (even those weighted by the inverse of popular measures, or randomly generated) have earned higher returns and higher Sharpe ratios than a cap-weighted benchmark, gross of costs. The authors’ interpretation was that high price predicts poor returns (as the well-known value and size factors seem to suggest) and therefore all portfolios that rebalance to prevent price determining weight will tend to outperform. Many “smart beta” managers, the authors imply, inadvertently harness and rely on this ubiquitous underlying price effect, while claiming a broader range of advantages for their products. Portfolio Rebalancing – Common Misconceptions 11 Exhibit A2 ‒ Contemporaneous Correlation Between Volatility Change and Excess Return, 1975-2015 EQUITIES Correlation 1.0 BONDS COMMODITIES TIPS 0.5 0.0 -0.5 20-day window US 10y Wheat UK 10y Soybeans Silver Natural Gas Gold Heating Oil Crude Corn Copper BCOM Index UK 10y GSCI Index Germany 10y US 10y Japan 10y Australia Hong Kong Sweden Italy Norway Spain Netherlands France Germany Switzerland UK Japan Canada RU2K NASDAQ S&P 500 Dow Jones -1.0 60-day window Source AQR. Hypothetical strategies based on daily returns from 1975 where available; some series start later. Returns are excess of cash. Volatility changes are based on rolling volatilities with 20- or 60-day window to match change window. Hypothetical data have inherent limitations, some of which are disclosed in the appendix. That analysis was gross of costs. In a world of transaction costs and capacity constraints, tilts away from the market portfolio do not come for free. All non-cap-weighted portfolios require periodic rebalancing and so incur higher transaction costs. Some are illiquid or impractical for institutions because they give large weights to small-cap stocks. Therefore we believe that smart beta investors should seek out those tilts – or factors – with the highest expected net returns. Long/short evidence unambiguously supports the outperformance of factors such as value, momentum and quality (and not their inverses!), all of which are also supported by economic intuition.20 Research on market frictions suggests that these factors are sufficiently robust and attractive to survive real-world costs.21 Could Risk-Targeting Exacerbate Market Volatility? Could risk-targeting processes create feedback loops of selling during a crisis, similar to the effect of portfolio insurance in 1987? Risk-targeting trades have a momentum bias if increases in volatility tend to coincide with negative returns. Exhibit A2 shows contemporaneous correlations between volatility changes and return for 34 assets and for two different horizons. Increasing volatility does indeed coincide with lower returns for equities, but for other asset classes the relationship is much weaker. For commodities, spikes in volatility are just as likely to be associated with price increases (risk-based rebalancing is actually contrarian during such episodes). Even in equities, risk-targeted capital currently represents such a small proportion of the total investor base that traditional stop-loss selling and nonquantitative increases in risk aversion are likely to be responsible for far more of any crisis selling. The rebalancing of risk-targeted long/short strategies is more complex. For these strategies, directional market shocks do not necessarily generate net buy or sell adjustments. Strategy losses and increased volatility may cause reductions of both long and short positions and such deleveraging can result in further losses for similar strategies (as in the “quant crisis” of August 2007), but any impact from – or effect on – directional market moves is likely to be mitigated by the offsetting nature of long/short positions and trades. 20See for example Asness, Moskowitz and Pedersen (2013) and Asness, Frazzini and Pedersen (2013). The findings of Arnott et al do not contradict this evidence. They do not apply factor tilts and reversed tilts, but rather test entirely price-agnostic weighting schemes and the inverses of those schemes, without regard to implementability. Alternative portfolio construction choices were explored by Amenc, Goltz and Lodh (2016), who found that factor-tilted portfolios did indeed outperform their inverses. 21Frazzini, Israel and Moskowitz (2012). 12 Portfolio Rebalancing – Common Misconceptions References Amenc, Noel, F. Goltz and A. Lodh, 2016, “Smart Beta is Not Monkey Business,” Journal of Index Investing, 6(4), 12-29. Andersen, Robert M., S. W. Bianchi and L. R. Goldberg, 2014, “Determinants of Levered Portfolio Performance,” Financial Analysts Journal, 70(5). Ang, Andrew, 2014, “Asset Management: A Systematic Approach to Factor Investing,” OUP. Arnott, Robert D., J. Hsu, V. Kalesnik and P. Tindall, 2013, “The surprising alpha from Malkiel’s monkey and upside-down strategies,” The Journal of Portfolio Management, 39(4). Asness, Clifford S., T. J. Moskowitz and L. H. Pedersen, 2013, “Value and Momentum Everywhere,” The Journal of Finance 68(3), 929–985. Asness, Clifford S., A. Frazzini and L. H. Pedersen, 2014, “Quality Minus Junk,” working paper. Asness, Clifford S., B. T. Hood and J. J. Huss, 2015, “Determinants of Levered Portfolio Performance: A Comment,” Financial Analysts Journal, 71(5). Asness, Clifford S., A. Ilmanen and T. Maloney, 2016, “Market Timing: Sin a Little,” AQR Whitepaper. Azuelos, Valerie, and R. Yasenchak, 2014, “Volatility and Rebalancing as a Source of Return,” white paper. Bernstein, William J., and D. Wilkinson, 1997, “Diversification, rebalancing, and the geometric mean frontier.” Booth, David G., and E. F. Fama, 1992, “Diversification returns and asset contributions,” Financial Analysts Journal, 48(3). Chambers, Donald, and J. Zdanowicz, 2014, “The limitations of diversification return,” The Journal of Portfolio Management, 40(4). Cheng, Minder, and A. Madhavan, 2009, “Dynamics of leveraged and inverse ETFs.” Engle, Robert F., 1982, “Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation,” Econometrica 50(4), 987–1008. Erb, Claude B., and C. R. Harvey, 2006, “The strategic and tactical value of commodity futures,” Financial Analysts Journal, 62(2). Frazzini, Andrea, R. Israel and T. J. Moskowitz, 2012, “Trading Costs of Asset Pricing Anomalies,” working paper. Hallerbach, Winfried G., 2014, “Disentangling Rebalancing Return,” Journal of Asset Management, 15, 301-316. Harvey, Campbell R., N. Granger, D. Greenig, S. Rattray and D. Zou, 2014, “Rebalancing Risk,” working paper. Hillion, Pierre, 2016, “The Ex-Ante Rebalancing Premium,” working paper. Hughson, Eric, M. Stutzer and C. Yung, 2006. “The misuse of expected returns,” Financial Analysts Journal, 62(6). Ilmanen, Antti, 2011, “Expected Returns,” Wiley. Ilmanen, Antti, and T. Maloney, 2015, “Portfolio Rebalancing: Strategic Asset Allocation’” AQR white paper. Jacquier, Eric, A. Kane, and A. J. Marcus, 2003, “Geometric or arithmetic mean: a reconsideration,” Financial Analysts Journal, 59(6). Moreira, Alan, and T. Muir, 2016, “Volatility Managed Portfolios,” working paper. Perchet, Romain, Raul Leote de Carvalho, Thomas Heckel and Pierre Moulin, 2014, “Inter-temporal Risk Parity: A constant volatility framework for equities and other asset classes,” working paper. Portfolio Rebalancing – Common Misconceptions 13 Perold, Andre F., and W. F. Sharpe, 1988, “Dynamic Strategies for Asset Allocation,” Financial Analysts Journal, 44(1). Qian, Edward, 2012, “Diversification return and leveraged portfolios,” The Journal of Portfolio Management, 38(4). Qian, Edward, 2014, “To Rebalance or Not to Rebalance: A Statistical Comparison of Terminal Wealth of FixedWeight and Buy-and-Hold Portfolios,” working paper. Willenbrock, Scott, 2010, “Diversification return, portfolio rebalancing and the commodity return puzzle,” Financial Analysts Journal, 67(4). Wise, A. J., 1996, “The Investment Return from a Portfolio with a Dynamic Rebalancing Policy,” British Actuarial Journal, 2(4). 14 Notes Portfolio Rebalancing – Common Misconceptions Notes Portfolio Rebalancing – Common Misconceptions 15 16 Notes Portfolio Rebalancing – Common Misconceptions Portfolio Rebalancing – Common Misconceptions 17 Disclosures The information set forth has been obtained or derived from sources believed by the authors and AQR Capital Management, LLC (“AQR”) to be reliable. However the authors and AQR do not make any representation or warranty, express or implied, as to the information’s accuracy or completeness, nor does AQR recommend that the attached information serve as the basis of any investment decision. This document has been provided to you for information purposes and does not constitute an offer or solicitation of an offer, or any advice or recommendation, to purchase any securities or other financial instruments, and may not be construed as such. This document is intended exclusively for the use of the person to whom it has been delivered by AQR and it is not to be reproduced or redistributed to any other person. This document is subject to further review and revision. This document has been prepared solely for information purposes. The information contained herein is only current as of the date indicated, and may be superseded by subsequent market events or for other reasons. Charts and graphs provided herein are for illustrative purposes only. Nothing contained herein constitutes investment, legal tax or other advice nor is it to be relied on in making an investment or other decision. There can be no assurance that an investment strategy will be successful. Historic market trends are not reliable indicators of actual future market behavior or future performance of any particular investment which may differ materially, and should not be relied upon as such. The information in this document may contain projections or other forward-looking statements regarding future events, targets, forecasts or expectations regarding the strategies described herein, and is only current as of the date indicated. There is no assurance that such events or targets will be achieved, and may be significantly different from that shown here. The information in this presentation, including statements concerning financial market trends, is based on current market conditions, which will fluctuate and may be superseded by subsequent market events or for other reasons. Performance of all cited indices is calculated on a total return basis with dividends reinvested. The indices do not include any expenses, fees or charges and are unmanaged and should not be considered investments. The investment strategy and themes discussed herein may be unsuitable for investors depending on their specific investment objectives and financial situation. Please note that changes in the rate of exchange of a currency may affect the value, price or income of an investment adversely. Neither AQR nor the authors assumes any duty to, nor undertakes to update forward looking statements. No representation or warranty, express or implied, is made or given by or on behalf of AQR, the authors or any other person as to the accuracy and completeness or fairness of the information contained in this presentation, and no responsibility or liability is accepted for any such information. By accepting this document in its entirety, the recipient acknowledges its understanding and acceptance of the foregoing statement. Diversification does not eliminate the risk of experiencing investment losses. There is no guarantee, express or implied, that long-term return and/or volatility targets will be achieved. Realized returns and/or volatility may come in higher or lower than expected. PAST PERFORMANCE IS NOT AN INDICATION OF FUTURE PERFORMANCE. Simulated l performance results (e.g., quantitative backtests) have many inherent limitations, some of which, but not all, are described herein. No representation is being made that any fund or account will or is likely to achieve profits or losses similar to those shown herein. In fact, there are frequently sharp differences between hypothetical performance results and the actual results subsequently realized by any particular trading program. One of the limitations of simulated results is that they are generally prepared with the benefit of hindsight. In addition, simulated trading does not involve financial risk, and no hypothetical trading record can completely account for the impact of financial risk in actual trading. For example, the ability to withstand losses or adhere to a particular trading program in spite of trading losses are material points which can adversely affect actual trading results. The simulated results contained herein represent the application of the quantitative models as currently in effect on the date first written above and there can be no assurance that the models will remain the same in the future or that an application of the current models in the future will produce similar results because the relevant market and economic conditions that prevailed during the hypothetical performance period will not necessarily recur. There are numerous other factors related to the markets in general or to the implementation of any specific trading program which cannot be fully accounted for in the preparation of hypothetical performance results, all of which can adversely affect actual trading results. Discounting factors may be applied to reduce suspected anomalies. This backtest’s return, for this period, may vary depending on the date it is run. Simulated performance results are presented for illustrative purposes only. Gross performance results do not reflect the deduction of investment advisory fees, which would reduce an investor’s actual return. For example, assume that $1 million is invested in an account with the Firm, and this account achieves a 10% compounded annualized return, gross of fees, for five years. At the end of five years that account would grow to $1,610,510 before the deduction of management fees. Assuming management fees of 1.00% per year are deducted monthly from the account, the value of the account at the end of five years would be $1,532,886 and the annualized rate of return would be 8.92%. For a ten-year period, the ending dollar values before and after fees would be $2,593,742 and $2,349,739, respectively. AQR’s asset based fees may range up to 2.85% of assets under management, and are generally billed monthly or quarterly at the commencement of the calendar month or quarter during which AQR will perform the services to which the fees relate. Where applicable, performance fees are generally equal to 20% of net realized and unrealized profits each year, after restoration of any losses carried forward from prior years. In addition, AQR funds incur expenses (including start-up, legal, accounting, audit, administrative and regulatory expenses) and may have redemption or withdrawal charges up to 2% based on gross redemption or withdrawal proceeds. Please refer to AQR’s ADV Part 2A for more information on fees. Consultants supplied with gross results are to use this data in accordance with SEC, CFTC, NFA or the applicable jurisdiction’s guidelines. There is a risk of substantial loss associated with trading commodities, futures, options, derivatives and other financial instruments. Before trading, investors should carefully consider their financial position and risk tolerance to determine if the proposed trading style is appropriate. Investors should realize that when trading futures, commodities, options, derivatives and other financial instruments one could lose the full balance of their account. It is also possible to lose more than the initial deposit when trading derivatives or using leverage. All funds committed to such a trading strategy should be purely risk capital. AQR Capital Management, LLC Two Greenwich Plaza, Greenwich, CT 06830 p: +1.203.742.3600 I f: +1.203.742.3100 I w: aqr.com